>Let (f,X) and (y,Y) be compactifications of S.>Assume h in C(Y,X) and f = hg. >>Thue h is a continuous surjection and when Y is Hausdorff>a closed quotient map.>>k = h|g(S):g(S) -> f(S) is a continuous bijection.>It it a homeomorphism? If so, what's a proof like?

I doubt that it follows that k is a homeomorphism,although I'm not going to try to give a counterexample.

Assuming it doesn't follow, then k being a homeomorphism"should" be part of the _definition_ of the partialorder on the class of compactifications that we'restruggling to define here.